Asymptotic Goldbach conjecture
Prove the asymptotic version of Goldbach's conjecture concerning representations of sufficiently large even integers as sums of two primes.
References
Assuming the asymptotic version of Goldbach's conjecture, Granville Theorem 2 showed that there is a subset $A\subseteq P$ with $|A\cap [1,x]| \asymp (x\log x){1/2}$ such that every large even integer is the sum of two elements of $A$.
Assuming the asymptotic version of Goldbach's conjecture, Granville Theorem 2 showed that there is a subset $A\subseteq P$ with $|A\cap [1,x]| \asymp (x\log x){1/2}$ such that every large even integer is the sum of two elements of $A$.
We assumed some conditions on the basis of our observations. With one of these assumptions we could be able to produce a function $\tau(n)$ and prove that $0<T(n)<\tau(n)<1-\frac{2}{n}$ for large $x$ which implies $x$ can be expressed as a sum of two odd primes. The other one leads to simply $0<T(n)<1$. All we need is to prove these conditions in future to complete the solution.
Though this (heuristic) arguments justify the Hardy-Littlewood conjecture, this does not provide a proof for it as the discrete correction factor and its application on powers of $\left(1-\frac{1}{p}\right)$ (as in the last step) are not yet proved rigorously.